If two sets X and Y have the same cardinality, can we assume that the f: X->Y is one to one? If not, and all Im given is that they're the same size, how can I prove that the function is injective? Thanks.

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- Sep 13th 2011, 08:54 PM #1

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## Re: One to One

Are you given any information about $\displaystyle f$? Since $\displaystyle |X| = |Y|$, there /exists/ a bijective function between these sets. But there also exist functions which are neither surjective nor injective (assuming that they have more than one element).

Or maybe the assignment is to find such an $\displaystyle f$?

- Sep 13th 2011, 10:03 PM #6
## Re: One to One

Well, that's pretty important haha. It's true that a mapping $\displaystyle f:X\to Y$ where $\displaystyle |X|=|Y|<\infty$ is injective if and only if it's surjective. Does that help? (a proof can be found on my blog)